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760f90e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 | import OAI.Computability.UniqueGames.Games.FactorizationLemmas
import OAI.Computability.UniqueGames.Games.Variation
namespace OAI
section
/-! Exact variation decomposition for the two choices of revealed endpoint.
The support of the actual reveal profiles makes both branch identities valid,
including zero-probability revealed endpoints. -/
namespace UniqueGamesTheorem.Foundations.Repetition
open scoped BigOperators
noncomputable section
variable {S X Y : Type*} [Fintype S] [Fintype X] [Fintype Y]
[DecidableEq X] [DecidableEq Y]
def maskedJoint (p : S × (X × Y) → ℝ) : (S × (X ⊕ Y)) × (X × Y) → ℝ := fun z =>
(if z.1.2 = Sum.inl z.2.1 then p (z.1.1, z.2) / 2 else 0) +
(if z.1.2 = Sum.inr z.2.2 then p (z.1.1, z.2) / 2 else 0)
def maskedModel (μ : Games.FiniteDistribution (X × Y)) (p : S × (X × Y) → ℝ) :
(S × (X ⊕ Y)) × (X × Y) → ℝ := fun z =>
Information.firstMarginal (maskedJoint p) z.1 * (revealProfile μ z.1.2).weight z.2
def leftRevealModel (μ : Games.FiniteDistribution (X × Y)) (p : S × (X × Y) → ℝ) :
S × (X × Y) → ℝ := fun z =>
(∑ y, p (z.1, (z.2.1, y))) * (revealProfile μ (Sum.inl z.2.1)).weight z.2
def rightRevealModel (μ : Games.FiniteDistribution (X × Y)) (p : S × (X × Y) → ℝ) :
S × (X × Y) → ℝ := fun z =>
(∑ x, p (z.1, (x, z.2.2))) * (revealProfile μ (Sum.inr z.2.2)).weight z.2
theorem sum_maskedJoint (p : S × (X × Y) → ℝ) :
(∑ z, maskedJoint p z) = ∑ z, p z := by
classical
have hrow (s : S) (q : X × Y) :
(∑ r : X ⊕ Y, maskedJoint p ((s,r),q)) = p (s,q) := by
simp [maskedJoint]
calc
_ = ∑ s : S, ∑ r : X ⊕ Y, ∑ q : X × Y, maskedJoint p ((s,r),q) := by
simp only [Fintype.sum_prod_type]
_ = ∑ s : S, ∑ q : X × Y, p (s,q) := by
apply Finset.sum_congr rfl
intro s _
rw [Finset.sum_comm]
simp_rw [hrow]
_ = _ := (Fintype.sum_prod_type _).symm
theorem maskedJoint_isProbability_iff (p : S × (X × Y) → ℝ) :
Information.IsProbability (maskedJoint p) ↔ Information.IsProbability p := by
classical
constructor
· intro h
constructor
· intro z
have hp := h.1 ((z.1,Sum.inl z.2.1),z.2)
simp [maskedJoint] at hp
linarith
· rw [← sum_maskedJoint]
exact h.2
· intro h
constructor
· intro z
have hp : 0 ≤ p (z.1.1,z.2) / 2 := div_nonneg (h.1 _) (by norm_num)
dsimp [maskedJoint]
split_ifs <;> linarith
· rw [sum_maskedJoint]
exact h.2
omit [Fintype S] in
theorem maskedJoint_firstMarginal_inl (p : S × (X × Y) → ℝ) (s : S) (x : X) :
Information.firstMarginal (maskedJoint p) (s, Sum.inl x) =
(∑ y, p (s, (x, y))) / 2 := by
simp [Information.firstMarginal, maskedJoint, Fintype.sum_prod_type,
div_eq_mul_inv, Finset.sum_mul]
rw [Finset.sum_comm]
simp
omit [Fintype S] in
theorem maskedJoint_firstMarginal_inr (p : S × (X × Y) → ℝ) (s : S) (y : Y) :
Information.firstMarginal (maskedJoint p) (s, Sum.inr y) =
(∑ x, p (s, (x, y))) / 2 := by
simp [Information.firstMarginal, maskedJoint, Fintype.sum_prod_type,
div_eq_mul_inv, Finset.sum_mul]
private theorem abs_half_sub_half_mul (a b c : ℝ) :
|a / 2 - b / 2 * c| = |a - b * c| / 2 := by
rw [show a / 2 - b / 2 * c = (a - b * c) / 2 by ring, abs_div]
norm_num
omit [Fintype S] in
theorem abs_masked_inl (μ : Games.FiniteDistribution (X × Y))
(p : S × (X × Y) → ℝ) (s : S) (u x : X) (y : Y) :
|maskedJoint p ((s, Sum.inl u), (x, y)) - maskedModel μ p ((s, Sum.inl u), (x, y))| =
if x = u then |p (s, (x, y)) - leftRevealModel μ p (s, (x, y))| / 2 else 0 := by
by_cases hx : x = u
· subst x
simp only [maskedModel, maskedJoint_firstMarginal_inl]
simpa [maskedJoint, leftRevealModel] using
abs_half_sub_half_mul (p (s, (u, y))) (∑ v, p (s, (u, v)))
((revealProfile μ (Sum.inl u)).weight (u, y))
· have hs := revealProfile_inl_support μ u (x, y) hx
simp [maskedJoint, maskedModel, hs, hx, Ne.symm hx]
omit [Fintype S] in
theorem abs_masked_inr (μ : Games.FiniteDistribution (X × Y))
(p : S × (X × Y) → ℝ) (s : S) (v : Y) (x : X) (y : Y) :
|maskedJoint p ((s, Sum.inr v), (x, y)) - maskedModel μ p ((s, Sum.inr v), (x, y))| =
if y = v then |p (s, (x, y)) - rightRevealModel μ p (s, (x, y))| / 2 else 0 := by
by_cases hy : y = v
· subst y
simp only [maskedModel, maskedJoint_firstMarginal_inr]
simpa [maskedJoint, rightRevealModel] using
abs_half_sub_half_mul (p (s, (x, v))) (∑ u, p (s, (u, v)))
((revealProfile μ (Sum.inr v)).weight (x, v))
· have hs := revealProfile_inr_support μ v (x, y) hy
simp [maskedJoint, maskedModel, hs, hy, Ne.symm hy]
theorem masked_totalVariation_identity (μ : Games.FiniteDistribution (X × Y))
(p : S × (X × Y) → ℝ) :
2 * Information.totalVariation (maskedJoint p) (maskedModel μ p) =
Information.totalVariation p (leftRevealModel μ p) +
Information.totalVariation p (rightRevealModel μ p) := by
have hleft (s : S) :
(∑ u : X, ∑ q : X × Y,
|maskedJoint p ((s, Sum.inl u), q) - maskedModel μ p ((s, Sum.inl u), q)|) =
(∑ q : X × Y, |p (s, q) - leftRevealModel μ p (s, q)|) / 2 := by
simp_rw [Fintype.sum_prod_type, abs_masked_inl]
simp [Finset.sum_ite_irrel, div_eq_mul_inv, Finset.sum_mul]
have hright (s : S) :
(∑ v : Y, ∑ q : X × Y,
|maskedJoint p ((s, Sum.inr v), q) - maskedModel μ p ((s, Sum.inr v), q)|) =
(∑ q : X × Y, |p (s, q) - rightRevealModel μ p (s, q)|) / 2 := by
have hcollapse :
(∑ v : Y, ∑ q : X × Y,
|maskedJoint p ((s, Sum.inr v), q) - maskedModel μ p ((s, Sum.inr v), q)|) =
∑ v : Y, ∑ x : X, |p (s, (x, v)) - rightRevealModel μ p (s, (x, v))| / 2 := by
simp [Fintype.sum_prod_type, abs_masked_inr]
rw [hcollapse, Finset.sum_comm]
simp [Fintype.sum_prod_type, div_eq_mul_inv, Finset.sum_mul]
have hmass :
(∑ z : (S × (X ⊕ Y)) × (X × Y), |maskedJoint p z - maskedModel μ p z|) =
((∑ z : S × (X × Y), |p z - leftRevealModel μ p z|) +
(∑ z : S × (X × Y), |p z - rightRevealModel μ p z|)) / 2 := by
calc
_ = ∑ s : S,
((∑ u : X, ∑ q : X × Y, |maskedJoint p ((s, Sum.inl u), q) - maskedModel μ p ((s, Sum.inl u), q)|) +
(∑ v : Y, ∑ q : X × Y, |maskedJoint p ((s, Sum.inr v), q) - maskedModel μ p ((s, Sum.inr v), q)|)) := by
simp only [Fintype.sum_prod_type, Fintype.sum_sum_type]
_ = ∑ s : S,
((∑ q : X × Y, |p (s, q) - leftRevealModel μ p (s, q)|) / 2 +
(∑ q : X × Y, |p (s, q) - rightRevealModel μ p (s, q)|) / 2) := by
simp_rw [hleft, hright]
_ = _ := by
simp [Fintype.sum_prod_type, div_eq_mul_inv, Finset.sum_add_distrib, Finset.sum_mul, add_mul]
unfold Information.totalVariation
rw [hmass]
ring
theorem leftRevealModel_totalVariation_le (μ : Games.FiniteDistribution (X × Y))
(p : S × (X × Y) → ℝ) :
Information.totalVariation p (leftRevealModel μ p) ≤
2 * Information.totalVariation (maskedJoint p) (maskedModel μ p) := by
rw [masked_totalVariation_identity]
linarith [Information.totalVariation_nonneg p (rightRevealModel μ p)]
theorem rightRevealModel_totalVariation_le (μ : Games.FiniteDistribution (X × Y))
(p : S × (X × Y) → ℝ) :
Information.totalVariation p (rightRevealModel μ p) ≤
2 * Information.totalVariation (maskedJoint p) (maskedModel μ p) := by
rw [masked_totalVariation_identity]
linarith [Information.totalVariation_nonneg p (leftRevealModel μ p)]
end
end UniqueGamesTheorem.Foundations.Repetition
end
end OAI
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